Two functions with C^n continuity over an interval:Which one's the smoothest?

• Jul 6th 2012, 01:13 PM
x3bnm
Two functions with C^n continuity over an interval:Which one's the smoothest?
According to the definition of Smooth function - Wikipedia, the free encyclopedia:

Functions that have derivatives of all orders are called smooth.

Suppose I've two different functions $\displaystyle f(x) = 2x^2$ and $\displaystyle g(x) = 4x^2$ of continuity $\displaystyle C^n$ over an interval $\displaystyle [0,4]$. Because both of these functions has derivatives of all order these two functions are smooth.

What I like to know is which one of these two are smoothest?

Is it possible to kindly help me detect which function($\displaystyle f(x)$ or $\displaystyle g(x)$) is the smoothest over the given interval?
• Jul 7th 2012, 01:16 PM
HallsofIvy
Re: Two functions with C^n continuity over an interval:Which one's the smoothest?
How are you defining "smoothest"? Some times texts will say that if f is differentable "n" times and g is differentiable "m" times, with n> m, then f is "smoother" than g. But you are using "smooth" to mean what those texts would call "infinitely smooth" so that does not apply here.
• Jul 8th 2012, 02:30 PM
x3bnm
Re: Two functions with C^n continuity over an interval:Which one's the smoothest?
Quote:

Originally Posted by HallsofIvy
How are you defining "smoothest"? Some times texts will say that if f is differentable "n" times and g is differentiable "m" times, with n> m, then f is "smoother" than g. But you are using "smooth" to mean what those texts would call "infinitely smooth" so that does not apply here.

Thanks HallsofIvy for answering my question. Yes I meant $\displaystyle f(x) = x^3$ is smoother than $\displaystyle g(x) = x^2$ because $\displaystyle x^3$ is one time more differentiable than $\displaystyle x^2$. I found the answer. Originally my problem was like this:

Suppose I've a acceleration vs. time graph where $\displaystyle x$ means time in millisecond and $\displaystyle y$ means acceleration in $\displaystyle \text{inch/ms}^2$

I've 2 sets of data with 4 points in each set. I've to find the approximated curve in each set that is smoother than the other(means one is more differentiable than other).

I found the solution by the way. All I have to do is curve fit in each set using Linear Algebra. Thanks for helping me see the way.